Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{5k^2 - 405}{-7k^3 - 70k^2 - 63k}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {5(k^2 - 81)} {-7k(k^2 + 10k + 9)} $ $ z = -\dfrac{5}{7k} \cdot \dfrac{k^2 - 81}{k^2 + 10k + 9} $ Next factor the numerator and denominator. $ z = - \dfrac{5}{7k} \cdot \dfrac{(k + 9)(k - 9)}{(k + 9)(k + 1)}$ Assuming $k \neq -9$ , we can cancel the $k + 9$ $ z = - \dfrac{5}{7k} \cdot \dfrac{k - 9}{k + 1}$ Therefore: $ z = \dfrac{ -5(k - 9)}{ 7k(k + 1)}$, $k \neq -9$